THE SUNYAEV-ZEL'DOVICH EFFECT IN ABELL 370

17 Pages·2007·0.31 MB·English

toappearinAstrophysicalJournalv538Aug1,2000
PreprinttypesetusingLATEXstyleemulateapjv.26/01/00
THE SUNYAEV-ZEL’DOVICH EFFECT IN ABELL 370
Laura Grego1, John E. Carlstrom2, Marshall K. Joy3, Erik D. Reese2, Gilbert P.
Holder2, Sandeep Patel4, Asantha R. Cooray2, and William L. Holzapfel5
to appear in Astrophysical Journal v538 Aug 1, 2000
ABSTRACT
We present interferometric measurements of the Sunyaev-Zel’dovich (SZ) effect towards the galaxy
0
cluster Abell 370. These measurements, which directly probe the pressure of the cluster’s gas, show the
0
gas distribution to be strongly aspherical, as do the x-ray and gravitational lensing observations. We
0
calculatethecluster’sgasmassfractionintwoways. WefirstcomparethegasmassderivedfromtheSZ
2
measurementstothelensing-derivedgravitationalmassnearthecriticallensingradius. Wealsocalculate
r
the gas mass fraction from the SZ data by deprojecting the three-dimensional gas density distribution
a
M and deriving the total mass under the assumption that the gas is in hydrostatic equilibrium (HSE). We
testtheassumptionsintheHSEmethodbycomparingthetotalclustermassimpliedbythetwomethods

6 andfindthattheyagreewithintheerrorsofthemeasurement. Wediscussthepossiblesystematicerrors
in the gas mass fraction measurement and the constraints it places on the matter density parameter,

1 Ω .
M
v
Subject headings: cosmic background radiation—cosmology: observations, galaxies:
5
clusters:individual(Abell 370)–techniques: interferometric
8
0
3 1. INTRODUCTION cm−3; and cools slowly (tcool > tHubble), mainly via ther-
0 mal Bremsstrahlung in the x-ray band. The x-ray sur-
0 Clustersofgalaxies,byvirtueofbeingthelargestknown
face brightness is proportional to the emission measure,
0 virialized objects, are important probes of large scale
S n2Λ(T )dl, where the integration is along the line
/ structure and can be used to test cosmological models. x ∝ e e
h of sight, and so, under simplifying assumptions, the gas
p Rich clusters are extremely massive, ∼ 1015M⊙, as indi- mass cRan be calculated from an x-ray image deprojection
cated by the presence of strongly gravitationally lensed
- andthemeasuredgastemperature. Sincethesoundcross-
o backgroundgalaxies and by the deep gravitationalpoten-
ingtimeoftheclustergasismuchlessthanthedynamical
r tialnecessarytoexplainboththe largevelocitydispersion
st (> 1000 km s−1) in the member galaxies and the high time,onemayreasonablyassumethat,intheabsenceofa
a recentmerger,theclustergasisrelaxedinthecluster’spo-
measured temperature (>5 keV) of the ionized intraclus-
: tential. The totalbinding mass canbe extractedfromthe
v ter gas. Dynamical mechanisms for segregating baryonic
gas density and temperature distribution under this as-
i matter fromdark matter onthese mass scalesaredifficult
X sumption. A significant body of work exists in which the
to reconcile with observations and standard cosmological
r models, and so within the virial radius the mass compo- gasmassfraction,fg,ismeasuredinthisway,withfg mea-
a surementsoutto radiiof1 Mpc ormore (White & Fabian
sition of clusters is expected to reflect the universal mass
1995; David et al. 1995; Neumann & Bohringer, 1997;
composition. Underthefairsamplehypothesis,acluster’s
Squires et al. 1997; Mohr, Mathiesen & Evrard 1999).
gasmassfraction,whichisalowerlimittotheitsbaryonic
The mean cluster gas mass fraction within approximately
massfraction,isthenalowerlimittotheuniversalbaryon
the virial radius was calculated in Mohr et al. (1999) to
fraction, i.e., f f .
gas ≤ B be (0.0749 0.0005)h−3/2. Here, and throughout the pa-
The luminous baryonic content of galaxy clusters is
±
per, we assume the value of the Hubble constant to be
mainly contained in the gaseous intracluster medium
(ICM).Thegasmassisnearlyanorderofmagnitudelarger H◦ =100h km s−1 Mpc−1.
Toderivethegasmassfractionfromx-rayimagingdata,
than the mass in optically observed galaxies (e.g., White
one is required to deproject the surface brightness into a
et al. 1993, Forman & Jones 1982). Hence, the gas mass
model for the density distribution. As the x-ray emis-
is not only a lower limit to the cluster’s baryonic mass, it
sion is proportional to the square of the gas density, the
is a reasonable estimate of it.
gas mass measurement can be biased by clumped, multi-
The intracluster medium has largely been studied
phase gas, should it be present. Also, the emission from
through observations of its x-ray emission. The ICM is
the cores of relaxed clusters may be dominated by cool-
hot, with electron temperatures, T , from 5 to 15 keV;
e
rarefied,withpeakelectronnumberdensitie∼sofn 10−3 ingflows,whichcomplicatethe interpretationofthex-ray
e
≃
1Harvard-SmithsonianCenter forAstrophysics,60GardenSt.,Cambridge,MA02138
2DepartmentofAstronomy&Astrophysics,5640S.EllisAve.,UniversityofChicago,Chicago,IL60637
3SpaceScienceLaboratory, ES84,NASAMarshallSpaceFlightCenter,Huntsville,AL35812
4DepartmentofAstronomy,UniversityofAlabama,Huntsville,AL35899
5DepartmentofPhysics,UniversityofCalifornia,Berkeley,CA94720
1
2
data and may bias the result strongly if not taken into gas mass fractions from interferometric SZ observations.
account (Allen 1998; Mohr et al. 1999). In addition, the Here, the shape parameters are derived directly from the
x-ray surface brightness is diminished in proportion to its SZdatasetratherthanfromanx-rayimage. Weapplythis
distance; S 1/(1+z)4, where z is the redshift of the methodtothe SZeffectmeasurementstowardsthecluster
x
∝
cluster, neglecting experiment-specific K-corrections, and Abell 370, which were made as part of an SZ survey of
so itbecomesincreasinglydifficult to makesensitivex-ray distantclusters,the firstresults of whichwerereportedin
measurementsoftheICMastheclusterredshiftincreases. Carlstrom, Joy & Grego (1996), and are further reported
We presentascheme for measuringthe gasmass fractions in Carlstrom et al. (1997). We choose this primarily be-
with the Sunyaev-Zel’dovicheffect which is different from causeithasbeenstudiedatopticalandx-raywavelengths,
the x-ray method in a number of ways, and also provides and so allows a comparison of the SZ data with other ob-
an independent measurement of f . servations. Upondetailedinvestigation,itisapparentthat
g
The Sunyaev-Zel’dovich(SZ) effect is a spectral distor- thisclusterisoneofthemostdifficultinoursampletoan-
tion of Cosmic Microwave Background (CMB) radiation alyze, with significant ellipticity and complicated optical
duetoscatteringofCMBphotonsbyhotplasma(Sunyaev and x-ray structure; as such, it serves as a test of the gas
& Zel’dovich 1970). The SZ effect can be detected signifi- mass fraction analysis method, which we plan to use on a
cantlyingalaxyclusters,wheretheionizedintraclustergas large sample of clusters.
serves as the scattering medium. A small fraction, 1%, With the interferometric SZ measurements, comple-
≤
of CMB photons are inverse-Compton scattered and, on mented by observationsat other wavelengths,we measure
average, gain energy. At frequencies less than about 218 the cluster’s gas mass fraction in two ways. First, we cal-
GHz, the intensity of the CMB radiation is diminished culate the surface gas mass from the SZ measurements
ascomparedto the unscatteredCMB,andthe SZeffectis and the surface total mass from strong gravitationallens-
manifestedasabrightnesstemperaturedecrementtowards ing observations and models. Second, we measure the gas
the cluster. This decrement, ∆T , has a magnitude pro- mass fraction in a manner similar to x-ray analyses. The
SZ
portional to the total number of scatterers, weighted by gas mass is inferred from a deprojected model; the total
their associatedtemperature, ∆TSZ n T dl, where n mass is determined from the spatial distribution of the
is the number density of electrToCnMs,BT∝is thee eelectron teme- gas under the isothermal hydrostatic equilibrium (HSE)
e R
perature, T is the temperature of the CMB, and the assumption. We test the assumptions made in the HSE
CMB
integration is again along the line of sight. Note that this analysis by comparing the total cluster masses derived in
is simply proportionalto the integrated electronpressure. the two methods.
Also,themagnitudeoftheSZdecrementisindependentof The optical and x-ray observations of this cluster are
redshift,soasthelongastheclusterisresolved(theexper- discussed in Section 2, and the SZ observations are dis-
iment’scharacteristicbeamsizeisnotlargerthantheangle cussed in Section 3. The method for modeling the SZ
subtended by the cluster), the SZ effect can be measured data is presented in Section 4, and the gas mass fraction
towards arbitrarily distant clusters. results and the systematic uncertainties are discussed in
A cluster’s gas mass is directly proportional to its in- Section 5. The cosmological implications of the results
tegrated SZ effect if the gas is isothermal. So under and plans for future work are discussed in Section 6.
the isothermality condition, an image deprojection is not 2. OPTICALANDX-RAYOBSERVATIONS
strictly requiredto obtain the gas mass. The cluster’s gas
The spatial distribution and redshifts of a number of
mass fraction can be calculated by comparing the inte-
Abell 370’s constituent galaxies have been measured op-
grated SZ decrement, in effect a surface gas mass, to the
tically. Mellier et al. (1988), using their 29 best galaxy
totalclustermassinthesamevolume. Thetotalmasscan
spectra, find Abell 370 has a mean redshift z = 0.374
bemeasuredwithstrongorweakgravitationallensing,for
and velocity dispersion σ = 1340+230 km s−1. These
example. The SZ images may also be deprojected to infer v −150
observations also show that the cluster is dominated by
the three-dimensional gas mass and the HSE mass. Since
two giant elliptical galaxies, at (α = 02h 39m 52.7s,
the SZ decrement is directly propotional to the electron J2000
δ = 01◦ 34′ 20.3′′) and (α = 02h 39m 53.2s,
density, the SZ image deprojection will not be affected J2000 J2000
δ = −01◦ 34′ 57.6′′), with a projected separation of
strongly by clumped gas. Thus, the cluster’s gas mass J2000
about 40′−′ in the north-south direction. The major axes
fraction can then be measured as a function of cluster ra-
of these two dominant galaxies are also oriented nearly
dius as well.
north–south. TheMellieretal.opticalisoplethmapshows
RecentclustergasmassfractionmeasurementsfromSZ
twodensitypeaks,withsimilarnorth-southorientation,al-
effect observations are presented in Myers et al. (1997).
thoughthesearenotcenteredonthetwoellipticalgalaxies.
In this work, the integrated SZ effect is measured using
The29galaxyvelocitiesdonotappeartobedistributedin
a single radio dish operating at centimeter wavelengths.
a Gaussian manner, although the statistics on this mod-
The integrated SZ effect is used to normalize a model for
est sample are not conclusive. Should this be significant,
the gas density from published x-ray analyses, and this
its may suggest that the cluster is bimodal or that it is
gas massis comparedto the published total massesto de-
not yet fully relaxed, and the assumption of a simple one-
termine the gas mass fraction. For three nearby clusters,
componentmodelinhydrostaticequilibriumwouldnotbe
A2142, A2256 and the Coma cluster, Myers et al. find
a gas mass fraction of (0.061 0.011)h−1 at radii of 1-1.5 appropriate. Themeanvelocitiesinthetwoisoplethpeaks
h−1 Mpc;fortheclusterAbel±l478,theyreportagasmass are not found to differ significantly, however, nor are the
fraction of (0.16 0.014)h−1. calculated barycenters of the two peaks, and so it is diffi-
± cult to make a firm conclusion on the state of the cluster
In this work, we describe a method to calculate cluster
based on these data alone.
3
Abell 370 was the first cluster found to have an asso- extended emission,andso may not be resolvedin this im-
ciated strongly gravitationally lensed arc (z = 0.725; age. It may also be explained by source variation; if the
arc
Soucail et al. 1988). In addition to this giant arc, which point source is an AGN, its surface brightness may have
apparently consists of multiple images of a single source, significantlyvariedbetweenthe1979IPCobservationand
threefaintpairsofimages,eachpairpresumedtobelensed the 1994-1995HRI observations. The image looks less el-
images of a single galaxy(Kneib et al. 1993),anda radial liptical than does the HRI image, but because the point
arc (Smail et al. 1996) have been found to be associated sourceemissioncannotbe excludedfromthe cluster emis-
with this cluster. Gravitational lens models have been sion in the IPC data, it is not useful for constraining the
used to map the totalmass distribution in the coreregion spatial distribution of the cluster gas.
of Abell 370. Kneib et al. (1993) find the position and Abell 370 has also been observed with the ASCA x-
elongation of the giant arc and the positions of the multi- ray satellite. Using 37.5 kiloseconds of ASCA GIS/SIS
ple images are best reproduced by a bimodal distribution observations, Mushotzky & Scharf (1997) determined the
of matter, with the two centers of mass nearly coincident emission-weighted average temperature of the cluster to
with the cluster’s two large elliptical galaxies. A subse- be7.13+1.05 keV.Analyzingthesamedata,Ota,Mitsuda,
−0.83
quently discoveredradialarcis consistentwith this model & Fukazawa (1998) find the emission-weighted tempera-
without modifications (Smail et al. 1996). ture to be 6.6+1.1 keV. Both of these values are system-
−0.9
Abell370wasobservedwiththeHighResolutionImager atically lower than the value obtained in the first analysis
(HRI) ofthe ROSATx-raysatellite. Inthe ROSATband- of these data by Bautz et al. (1994), who obtain a value
pass (0.1-2.4 keV), x-ray Bremsstrahlung emission from of 8.8 0.8 keV. The discrepancy is presumably because
a plasma at gas temperatures typical of clusters depends the Mu±shotzky et al.and Ota et al.analysesused the val-
onlyweaklyonthegastemperature. InFigure1,weshow ues of the response matrices of the ASCA telescope and
an HRI image of A370, the result of 30 kiloseconds x-raydetectorswhichhavebeenrefinedsincetheBautzet
∼
of observation time. The bright source to the north, at al. analysis (M. Bautz, private communication). For our
αJ2000 = 02h 39m 55s, δJ2000 = −01◦ 32′ 33′′, is coin- analysis,we adopt the Ota et al. value of 6.6+−10..19 keV, be-
cident with what appears in the Digital Sky Survey to cause this published analysis includes detailed discussion
be a nearby elliptical galaxy, NPM1G -01.0096 (Klemola, of the spectral fitting procedure.
Jones & Hanson 1987), and does not appear to be asso- The brightpointsourceevidentintheROSATHRI im-
ciated with A370. The image indicates that the intra- age has not been removed from the ASCA spectra in any
cluster medium also is elongatedin the north-southdirec- ofthe threepublishedanalyses. Ifthis emissionoriginates
tion, and the emission is strongly peaked at the position in Bremsstrahlung emission from hot gas in the ellipti-
of the southern dominant elliptical galaxy, numbered 35 calgalaxy,itscharacteristictemperatureisexpectedtobe
inMellieretal.(1988),andisoffsetfromthe centerofthe lessthan1or2keV(Sarazin&White 1988andreferences
largescaleemission. The clustergasdistributiondoes not therein). Since the sensitivity of ASCA is optimized for
appear clearly bimodal at this resolution. higher energies than this, we would not expect the mea-
We determine the x-ray surface brightness from the sured temperature to be greatly contaminated. We assess
ROSAT HRI image, which has been filtered to include whetherthesourceisextended,andfindthatitispointlike
only PHA channels 1-7 in orderto reduce the background attheresolutionofthe HRIwitha2.4σ certainty. Asthe
level (David et al. 1997) and blocked into 8′′ 8′′ pixels. source is unresolved, this emission is also consistent with
A circular region with a radius of 24′′ cent×ered on the a galactic cooling flow; if this were the case, the tempera-
pointlike x-ray source to the north of the cluster was ex- ture again shouldn’t be greatly contaminated. Should the
cluded from our modeling. We fit a β-model to the data; emission originate from an AGN in the galaxy, though,
this model described in more detail in Section 4, which the effect on the measured temperature is more difficult
parametrizes the x-ray surface brightness with a core ra- to predict, and will depend strongly on the index of the
dius,θc,andapowerlawindex,β. Thesurfacebrightness power law of the AGN spectrum.
of a spheroidally symmetric, isothermal gas distribution Withoutfurthermeasurements,wecannotruleoutthat
will have ellipticalcontours,andso we also fit for the axis the cluster’s spectrum is contaminated by emission from
ratio. an AGN. It is also possible that if the cluster is in
We find the best-fit axis ratio of the x-ray image to fact bimodal with two distinct subclusters, the measured
be 0.73 0.03, with major axis exactly north-south, and emission-weighted average T will be a value between the
the cent±er position to be αJ2000 = 02h 39m 53s, δJ2000 = temperaturesof the two subcelusters. Lackingmoredefini-
01◦ 34′ 33′′. The best fitting shape parameters for the tive measurements,we continue with ourassumptionthat
−elliptical model are β= 0.70±0.03 and θc= 85.0±3.5′′. If the gas is isothermal at kT =6.6+−10..19 keV. The estimated
the axis ratio is set to equal 1.0, the best fitting shape effectsofthedescribeduncertaintiesontheresultsaredis-
parameters are β= 0.72 0.03 , θc= (70.1 3.2)′′ (90% cussed in Section 5.3.
± ±
confidence on a single parameter). We compare A370’s measured line-of-sight velocity dis-
Abell 370 was also observed with the Einstein IPC for persion with our adopted gas temperature, and find
4000seconds. IntheIPCimage,thenorthernpointsource β µm σ2/kT = 1.87+1.02. If the energy per unit
is not apparent. This may be because the FWHM of the msapsecs c≡ontainpedvin tehe galaxi−e0s.5is9equal to that in the gas,
instrument pointspreadfunction is about90′′ atthe mid- and the cluster is spherical, we expect β to be 1. If
spec
dle of the 0.4-4.0 keV spectral range of the instrument. the temperature measurement is biased low, e.g., fr∼om an
(It improves to about 60′′ at higher energies.) The point AGN, this will contribute to the β discrepancy. Elon-
source is offset by less than 120′′ from the center of the gation of the cluster potential alonsgpetche line of sight or a
4
alignment of two (or more) subclusters will also enhance 0.725 0.07mJy. Thepointsourcemodelisremovedfrom
±
the measured β . theu-v orspatialfrequency,datasetinordertoconstruct
spec
Theopticalandx-rayobservationssuggestthatA370is the SZ decrementmap. A Gaussiantaper withFWHM of
not a relaxed cluster, and may be heavily substructured, 1.2kλ is appliedin theu-v plane to the combineddataset
but due to insufficient signal-to-noise ratios and contam- and map was CLEANed, restricting the CLEAN compo-
inating sources, these observations still leave some ambi- nents to the central200′′, about the size of the decrement
guity. in the primary. The map presented in Figure 2a is made
3. SZOBSERVATIONS witharestoringbeamofGaussianFWHM59′′ x86′′,with
contourintervalsof1.5σ. The rms noiseinthe mapis50
We observedAbell 370for 50hoursovernine days with µ Jy beam−1, or 14.9 µK, and the integrated flux density
the Owens Valley Radio Observatory (OVRO) Millimeter of the source is 1.54 0.17 mJy.
Arrayin 1996August and43hours oversix days with the We employed−nine±of the 6.1 meter telescopes of
Berkeley-Illinois-Maryland Association (BIMA) Millime- the BIMA array for SZ observations, using the same
terObservatoryin1997June-August. Wesetourpointing centimeter-wave receivers we used at OVRO. We center
center to αJ2000 =02h 39m 52.5s, δJ2000 = 01◦ 34′ 20′′, the 800 MHz bandwidth of the BIMA digital correlator
−
the position of the northern dominant elliptical galaxy. at 28.5 GHz. We achieved system temperatures of 30-55
We outfitted the millimeter telescopes with centimeter- K, scaled to above the atmosphere, depending on eleva-
wavelength receivers to broaden the instrument’s angu- tionandatmosphericwatercontent. Holographicantenna
lar resolution to the large angular scales typical of clus- pattern measurements were also made with this system.
ters. The receivers are based on low noise High Electron The primary beam at BIMA is nearly Gaussian with a
Mobility Transistor (HEMT) amplifiers (see Pospieszalski 396′′ FWHM. The data were edited and reduced using
1995), operating from 26 to 36 GHz. The single-mode re- the MIRIAD data reduction program (Sault, Teuben &
ceiversareconstructedtorespondonlytocircularlypolar- Wright 1995), taking care to the remove any spurious in-
ized light. The receivers interface with each array’s delay terferencefromthespectralchannelsanddroppingthelow
lines and correlator. The standard observational scheme signal-to-noise channels at the spectral filter edges.
entailed interleaving 20 minute cluster observations with Again using DIFMAP, we find the same point source
observations of a strong reference source near the clus- at about 45′′ to the east of the pointing center in a high
ter,whichallowsmonitoringoftheinstrumentalamplitude resolution (projected baselines longer than 1.4 kλ) image.
and phase gain. The amplitude gain drifts were minimal; ThepointsourcemeasuredatBIMAin1997hasfluxden-
thegainchangeswerelessthanapercentoveramanyhour sity of +0.70 0.17 mJy. The primary beam attenuation
track, and the average gains were quite stable from day at the point s±ource position in the BIMA system is 6%
to day. The amplitude gains were calibrated by compar- less than the attenuation at this position at OVRO∼, and
ing the measured flux of Mars with that predicted by the so the point source observations are consistent with the
Rudy model for Mars’ whole disk brightness temperature flux being constant in time. We model and subtract the
(Rudy, 1987) at the observed frequency and the apparent point source from the data set, apply a Gaussian taper
size of Mars as indicated by the Astronomical Almanac. with FWHM of 0.8 kλ in the u-v plane and construct a
The Rudy model is a radiative transfer model with an es- map of the decrement. The map was CLEANed, restrict-
timated accuracy of 4% (90% confidence) at centimeter ing the CLEAN components to the central 300′′ of the
wavelengths. Data taken when the projectedbaseline was image, about the size of the primary beam. ∼The restoring
within 3% of the shadowing limit were not used. beam used to make the map in Figure 2b. is a Gaussian
The OVRO Millimeter array consists of six 10.4 me- withFWHM 91′′ x95′′,andthe contourspresentedareat
ter telescopes. The OVROcontinuumcorrelatorwasused intervals of 1.5 σ. The rms noise in the map is 180 µJy
to correlate two 1 GHz bands centered at 28.5 GHz and beam−1,or31.4µK,andtheintegratedfluxdensityofthe
30 GHz. The system temperatures ranged from 40-60 K, source is 4.34 0.52 mJy.
depending on elevation and atmospheric water content. Itisins−tructiv±etocomparethegeneralcharacteristicsof
The primarybeams of the telescopes were measuredholo- theSZ imagestothex-raymap. Atthe resolutionofboth
graphically and can be approximated as Gaussian with a the OVRO and BIMA instruments, the gas is extended in
full width at half maximum (FWHM) of 252′′. The data the north-south direction, though less markedly than in
werecalibratedandeditedusingtheMMAdatareduction the x-raymap. The substructuresuggestedby the lensing
package (Scoville et al. 1993) taking care to remove data and velocity dispersion data is not evident here. We pro-
taken during poor weather or which show any anomalous ducemapsfromeachofthedatasets,increasingtheimage
phase jumps. resolution, but the gas still does not show significant sub-
We image the data using the DIFMAP package (Shep- structure orbimodality. The SZ decrementdistributionis
ard, Pearson,& Taylor 1994). Examining images of A370 not peaked at the pointing center, but south of it, at a
madewithprojectedbaselinesgreaterthan2.0kλ,wefind position about halfway between the two elliptical galaxies
apointsource45′′ totheeastofthemapcenter(mapcen- (this will be quantified in the next section). The detailed
ter is the pointing center) in both continuum bands. Its appearance of the SZ map, especially the shapes of the
measuredfluxdensityinthe28.5GHzband,attenuatedby least significant contours, depends on the specific method
the instrumental primary beam pattern, is +0.69 0.10 of point source removal and the CLEANing of the data,
±
mJy and is +0.84 0.10 mJy in the 30 GHz band. This andsoitisnotusefultocomparethespatiallyfilteredand
±
source, 0237-0147, is discussed further in Cooray et al. CLEANed SZ and x-ray images at such a level of detail.
1998. We combine the two continuum channels, and find Aswewilldiscussinthefollowingsection,thequantitative
thebestfittingmodelisapointsourcewithfluxdensityof
5
analysisoftheSZdataisdoneusingtheu-v datadirectly. If the cluster gas is isothermaland its symmetry axis is in
theplaneofthesky,thiselectrondensitydistributionleads
4. MODELING to the following two-dimensional SZ temperature decre-
In order to assess the data more quantitatively, we fit ment:
a parametrized model to the SZ brightness distribution. 1−3β
θ2 2 2
Since the SZ data are taken in the spatial frequency, or η
∆T (θ)=∆T(0) 1+ , (4)
u-v, domain, we do the model comparisons in u-v domain θc2!
as well.
We measure the SZ temperature decrement in units of where θ = η/D , D is the angular diameter distance,
η A A
antenna temperature, T , which relates an observed in- θ = r /D , and ∆T(0) is the temperature decrement at
a c c A
tensity change to a Rayleigh-Jeans temperature change, zero projected radius,
∆Iν = 2kcν22∆Ta. In order to recover the true blackbody 2 −23β
temperaturedecrement,the∆T measuredandreported l
RJ
hneortesstrhiocutllydibnetmheulRtiapylileedighb-yJeaanfascltiomrito.f 1T.h02e1S,Zastewmepaerre- ∆T(0)∝Tene◦Z 1+(cid:18)rc(cid:19) ! dl, (5)
ature decrement, ∆TSZ, is proportional to the Compton where the integral, dl is along the line of sight. Formally,
y-parameter, TCMB this integral extends from the observer along the line of
kσ sight through the cluster infinitely; in practice, a cutoff
T
y = n (l)T (l)dl, (1)
m c2 e e radius for the cluster is used. The β-model distribution
e Z of the electron density projects to a β-model distribution
wherek isBoltzmann’sconstant,σ istheThomsonscat-
T ofthe SZ decrementif the systemobeys sphericalorellip-
teringcrosssection,m istheelectronmass,n istheelec-
e e soidal symmetry.
tron density, T is the electron gas temperature, and the
e We perform a χ2 analysis of the β-model, by compar-
integral extends along the line of sight (dl). The propor-
ingβ-modelstothecombinedBIMAandOVROdatasets.
tionality depends on the observing frequency; it depends
We vary the following parameters: centroid position, β,
also on the electron temperature when relativistic correc-
θ , axis ratio (defined to be < 1), position angle (defined
tions are included6 (Rephaeli 1995, Challinor & Lasenby c
counter clockwise from north), and ∆T(0). The position
1998).
and flux density of the radio-bright point source are also
At 28.5 GHz, ∆TRJ = 1.92 y in the non-relativistic
TCMB − fit. The fitting procedure is conducted in several steps.
Rayleigh-Jeans approximation, where we adopt the value
The model corresponding to each set of the seven clus-
oftheCMBofFixsenetal.(1996)derivedfromtheCOBE
ter fit parameters and the three point source parameters
FIRASmeasurements,T =2.728K.Includingtherel-
CMB is multiplied by the primary beam response. The Fourier
ativistic corrections for kTe =6.6 keV, T∆CTMSZB =−1.86 y. transform of the result is compared directly with the in-
We fit a simple model to the SZ data. The β-model terferometerdata. Weusethe holographicallydetermined
(Cavaliere&Fusco-Femiano1976,1978)isfrequentlyused primary beams when modeling the data, and the entire
to fit the density profiles of galaxy clusters. The spheri- datasets are used to do the analysis. The inner u-v ra-
callysymmetric,isothermalβ-modeldescribesne,theden- diuscutoffisdeterminedbythe shadowinglimit,the limit
sity distribution of the gas, as varying as a function of where one telescope would partially block another, i.e.,
cluster radius, r: when the projected baseline is less than the diameter of a
r2 −3β/2 telescope dish. For the BIMA data this limit is 0.58 kλ
ne(r)=ne◦ 1+ r2 , (2) and for the OVRO data it is 1 kλ. The χ2 statistic is
(cid:18) c(cid:19) minimized, and the best fit values are determined using a
wherene◦ isthenumberdensityofthegasatthecenterof downhill simplex method.
the spheroidally symmetric gas distribution; rc, the core Performing the fitting procedure, we find the best fit
radius, is a characteristic size of the cluster; and β is the parametervaluesforA370are: θ =93′′,β =1.77,∆T(0)
c
power law index. = 609µK,andaxisratio=0.64withmajoraxisexactly
Sincetheclusterappearsellipticalinprojection,wegen- no−rth-south. The best fit central position is 20′′ to the
eralize the spherically-symmetric β-model and allow the south of the pointing center, at α = 02h 39m 53.2s,
J2000
gas density profileto be spheroidallysymmetric, i.e., with δ = 01◦ 34′ 40.4′′, about halfway between the two
J2000
biaxial symmetry. In the spheroidally symmetric model, giantellip−ticalgalaxies,and7′′ northofthefittedcentroid
theelectronnumberdensityinaprolatespheroidisafunc- forthex-rayimage. Theχ2 statisticforthebestfitvalues
tion of η2 = r2 +a2z2, where r and z are the radial and is135798for136136degreesoffreedom,yieldingareduced
height coordinates in the cylindrical coordinate system, χ2 of 0.9975. As points of comparison, we fit to the data
and a < 1 is the axis ratio of the two unique axes. If anullmodelforthe cluster (withthe pointsourcecompo-
the spheroid is oblate, then the density is a function of nent), which gives a reduced χ2 of 0.9986; we also fit the
η2 = a2r2 +z2, with a < 1. The electron density then cluster to a negative point source, finding a reduced χ2 of
follows the distribution: 0.9986.
η2 −3β/2 In order to determine the uncertainties in these fitted
ne(η)=ne◦ 1+ r2 , (3) parameters, the χ2 statistic is derived for a large range
(cid:18) c(cid:19)
6The change in spectral intensity due to the Sunyaev-Zel’dovich effect is calculated in Equation (4-8) Challinor & Lasenby (1998):
∆(TSZ/T)RJ = (eyxx−2e1x)2[xcoth(x/2)−4+θef(x)],wherex= khTνe andθe= mkeTce2. Thelasttermcorrectsforrelativisticeffects. At28.5GHz,
f(x)=3.58.
6
of β, θ , and ∆T(0), keeping centroid position, position lowing the axis ratios of the two components to vary does
c
angle, andpoint sourceposition andflux fixed to the best not improve the fit.
fit values. Presented in Figure 3a is a contour plot of the
β and θ fit results, where ∆T(0) is allowed to assume its 5. ANALYSIS
c
best fit value at every pair of β and θ . The axis ratio is
c 5.1. Comparison of SZ Gas Mass to the Strong Lensing
left to assume its best fit value at each set of β, θ , and
c Mass Estimate
∆T(0); it varies from 0.60 to 0.68 for points with ∆χ2 <
5 fromthe best fit point. The centroidposition, when left As indicated by Equation 1, the SZ brightness at any
to assume its best fit value at each point, does not ap- point in the two-dimensional projected image is simply
preciably change, varying less than 5′′. Figure 3b shows proportional to the integrated electron density along the
the contours derived when fitting the data to a spherical lineofsight,ifthegasisisothermal. Undertheisothermal
model for the gas distribution, i.e., the axis ratio is fixed assumption, we can directly measure the total number of
toavalueof1.0;inthiscase,thebestfitparametervalues electrons in the gas contained in the cylindrical volume of
are θ = 34′′, β = 0.86, ∆T(0) = 785 µK, the reduced a chosen radius, the long axis of which is defined by the
c
χ2 statistic was 0.9976. − line of sight. The total mass in the ionized phase can be
The full line contoursare markedfor ∆χ2 = 2.3,4.61, calculatedfromthisassumingavalueforthenumberofnu-
and 6.17 which indicate 68.3%, 90.0%, and 95.4% confi- cleonsperelectron. Ifthegashassolarmetallicity,asmea-
dence, respectively,for the two-parameterfit. The dashed sured by Anders & Grevesse (1989), the nucleon/electron
lines indicate ∆χ2 = 1.0, 2.71, and 6.63; the projection ratiois1.16. Thenucleon/electronratiochangeslessthan
onto the β or θ axis of the interval contained by these 1% for values of the metallicity from 0.1 to 1.0.
c
contours indicate the 68.3%, 90% and 99% confidence in- The interferometric measurements recover much of the
terval on the single parameter. The values of ∆T(0) for total SZ decrement on the angular scales measured, i.e.
which∆χ2 <1ateach(β,θ )pointrangeabout15%from the integrated flux from the data comprises 40-50% (de-
c
the best fitvalue. This illustratesthat β and θ arecorre- pendingontheinstrument)oftheintegratedfluxinabest
c
lated strongly and are not individually well constrained fitmodeltothesedata. Andsotheunmodeledinterferom-
by these data. This poor constraint is at least partly eter data provides a strong lower limit to the integrated
due tothe lowdeclinationofA370,whichmakessampling SZdecrementatthe angularscalesofinterest. The model
non-redundant spatial frequencies difficult at OVRO and is fitted to the u-v data in order to estimate the full, two-
BIMA. The shape parameters fit from the x-ray data are dimensional decrement, and thence the surface gas mass.
notconsistentwiththosefromthe SZdata. This couldbe This method does not assumeanything aboutthe state of
a result of shocks and complexity in the gas phase which the ICM other than that it is isothermal.
affect the x-ray emission and SZ effect differently. This We fit the SZ data in the three-dimensional parameter
discrepancy is difficult to resolve with the information at space 0.4 <β < 4.0,10′′ <rc <250′′, 2000µK <∆T(0)
−
hand;forthepurposeofthenextsection,investigatingthe < 200 µK, both for elliptical and circular models. At
−
gasmassfractionofthe clusterusingthe SZ effect,weuse each β, θc, and ∆T(0) point, we calculate the surface gas
the SZ fitted parameters only. massandtheχ2 statistic,fromwhichconfidencelimitsfor
We also fit the data with a β-model which includes a the surface gas mass are determined. Although the 68%
truncation of the gas distribution at a given radius. With confidence region contains a large range of β and θc val-
truncationradiifrom300′′ to1000′′,theshapeparameters ueswithin68%confidence,aswasevidentinFigure3,the
(β, θ , ∆T(0)) best fit values do not change appreciably, gas mass, which depends on all three gridded parameters,
c
much less than the variation contained within the ∆χ2 < is constrained relatively well. In Figure 4, we show the
1 region. The χ2 statistic changes only minimally for dif- derived surface gas mass at radius 65′′ as it varies with β
ferent cutoff radii, with ∆χ2 less than 0.1. This indicates and θc in the circular β-model fitting. The gas mass in
that the possible systematic uncertainty in the fitted pa- this geometry is 1.5+−00..76×1013h−2M⊙. The temperature
rameters due to modeling the data without a cutoff is not decrement is allowed to assume its best fit value at each
significant. point for this figure, although for the quantitative analy-
Since the optical and x-ray data suggest that Abell 370 sis, the full range of ∆T(0) is used. The isomass surfaces,
mayhaveabimodalgravitationalpotential,wealsofitthe shown in greyscale, follow the shape of the confidence in-
data with a pair of circular β-models, each allowed inde- terval contours, indicating that sets of parameters which
pendent shape parametersandposition. The best fit two- fitthe datawellwillpredictthe samegasmass. This isto
component model has one component centered at 50′′ be expected, since under the isothermal assumption, the
south of the pointing center,10′′ southof the larges∼outh- SZ flux is directly proportional to the gas mass, and the
ern elliptical galaxy; the best fit position of the second fit parameters must reproduce the same observed flux at
component is 5′′ north of the pointing center, i.e., 5′′ the angular scales where the flux is best measured.
north of the la∼rge northern elliptical. The southern com- We derive a surface gas mass to compare with the lens-
ponent has best-fit parameters β = 0.65, θ = 10′′, and ing mass using our fitted model parameters. The lensed
∆T(0) = 765 µK; the northern componentchas best-fit arc in Abell 370 has a radius of curvature of about 30′′
parameter−s β = 0.83, θ = 27′′, and ∆T(0) = 605 µK. centered nearly halfway between the dominant galaxies;
c
Adding a second component reduces the χ2 stat−istic by 2 this center is at about the same position of the gas den-
when 3 new parameters are introduced (compared to the sity centroid in the one component β model. We calcu-
singleβ-modelwithaxisratioandpositionanglefree),and late the surface gas mass in a cylindrical volume centered
therefore is not a significantly better fit to the data. Al- at this position and with an elliptical (axis ratio = 0.64)
cross-section, using the fits to the elliptical β-model. We
7
choose a major radius of 40′′ since the lens model should Thesecalculationsrequireanassumptionaboutthegeom-
be most accurate near the lensed arc’s radius. The SZ- etry of the cluster, since the core radius and extent of the
derivedsurfacegasmasswiththeellipticalcross-sectionis clusteralongthelineofsightarenotknown. Foranoblate
5.4+−11..20×1012h−2M⊙, at 68% confidence. Included in the ellipsoid, the axis of symmetry is the cluster’s minor axis,
error estimates are the uncertainty due to the fit parame- andthecoreradiusintheline-of-sightdirectionisequalto
ters, the SZ absolute calibration uncertainty, and the gas the cluster’s observed major axis; for a prolate ellipsoid,
temperaturemeasurementuncertainty. Theuncertaintyis theaxisofsymmetryisthemajoraxis,andthecoreradius
not appreciably smaller if we restrict the shape parame- in the line-of-sight direction is equal to the minor axis.
terstothosewhichtypicallyfoundinx-rayimageanalyses The fit parameters are then used to constrain the total
of clusters, i.e., β < 1.5. The uncertainties of the masses mass. Hydrostatic equilibrium implies
derivedfromthe ellipticalfits aresmaller thanthose from
1
the spherical fits because the confidence contours follow
p = Φ, (6)
gas
the isomass contours more closely. ρ ∇ −∇
gas
Wealsoderivethegasmassinthemodelwithtwocom-
where ρ and p are the gas density and pressure, re-
ponents. The best fitting two-componentβ-modelparam- gas gas
spectively. The cluster’s gravitational potential, Φ, can
eters are integrated in a cylinder with radius 40′′, cen-
be related to the total mass density, ρ , by Poisson’s
tered on the centroid of the best fit single component grav
equation,
model. Thisyieldsagasmassof4.84 1012h−2M⊙,consis-
×
tent with the single component model. Since the bimodal
2Φ=4πGρ , (7)
modeldoublestheparameterspaceoverwhichfitsmustbe ∇ grav
made, making a comprehensive parameter fit unfeasable, where G is the gravitational constant. We can solve for
we approximate the statistical uncertainty from the fit to the density of the cluster’s gravitationalmass by combin-
be the same as in the single model fit, about 20%. ing Equations 6 and 7:
These surface gas masses are compared to the cluster’s
totalmassinthesamevolumeimpliedbythestronggravi- 1 1
ρ = p . (8)
tationallensingmeasurements. Thesurfacetotalmasscan grav −4πG∇· ρ ∇ gas
(cid:18) gas (cid:19)
be inferred from a model of the cluster mass distribution
We relate the pressure, density, and temperature of the
whichpredicts the observedgravitationallensing. We cal-
gas through the equation of state:
culate the gravitational mass using the lensing model in
Kneib et al. (1993) for the same volume, centered at the
ρ kT
SZ model fit center. Although the uncertainties on the p = gas e, (9)
gas
µm
model parameters are less than ten percent, we make a p
conservative estimate of 20% for the uncertainty, to al- where k is Boltzmann’s constant, µ is the mean molecu-
low for variations of this model. Comparing this mass, lar weight of the gas, and m is the proton’s mass. To
9.6+−11..99 ×1013h−1M⊙, to the single-component gas mass calculate µ, we again assume tphe gas has the solar metal-
yieldsagasmassfraction,f ,of(0.056+0.017)h−1;thetwo- licityofAndersandGrevesse(1989)andthatµisconstant
g −0.015
component modelyields a slightly lowergasmass fraction throughout the gas. Making the assumption that the gas
of (0.046 0.013)h−1. The cluster’s angulardiameter dis- is isothermal, we write Equation 8 in the form:
±
tance was calculated assuming Ω = 0.3 and Ω = 0. If
M Λ
ΩM =1.0,ΩΛ =0,theangulardiameterdistancetoA370 ρ = kTe 2lnρ . (10)
will be 6.5% smaller, as will fg. If ΩM = 0.3 and ΩΛ = grav −4πGµmp∇ gas
0.7, f will be 9% larger.
g ∼ Note that the gravitationalmass density depends only on
the shape of the gas distribution, and so is independent
5.2. Comparison of SZ Gas Mass to Hydrostatic,
of the value of the central gas density and the gas mass
Isothermal Mass Estimates
fraction. Using the derived shape parameters, β, θ , and
c
Sincestronggravitationallensinginclustersisrelatively projectedaxisratio,a,andthemeasuredgastemperature,
rare and is restricted to the cores of clusters, and weak weestimatethetotalmassdensityofthecluster. Weagain
lensing analyses are published for only selected clusters, choosethe simplest geometries,that of oblate andprolate
wealsoconsiderthemoregeneralmeansofcalculatingthe ellipsoids, and integrate the density within the same vol-
clustergasfraction,deprojectionofthedensitymodeland ume as we do the gas density.
the HSE assumption. We assume the gasis inhydrostatic Forthesamerangeofβ,θ ,and∆T(0)asweusedinthe
c
equilibrium, is isothermal, and is spheroidally symmetric; cylindricalgeometryanalysis,wecalculatethecluster’sel-
forsimplicity,weassumethesymmetryaxisisintheplane lipsoidal gas mass, HSE mass, and gas mass fraction for
of the sky. The surfaces of constant electron density are bothprolateandoblategeometries,andthe∆χ2 fromthe
then concentric ellipsoids. We then compare the gas mass best fit parameters. From ∆χ2 = 1 range of fit param-
and the HSE mass from the deprojected model to deter- eters, we derive the 68% confidence intervals for the gas
mine the gas mass fraction. mass and the mean gas mass fraction calculated within
Specifically, to determine the gas mass, we extract the different major axis radii.
centralelectrondensity,ne◦,fromthedeprojectedβ-model We prefer to measure the masses and mass fractions
and measuredelectron temperature by performing the in- in the largest volume permitted by our method, since
tegralalongthelineofsightinEquation5andintegrating the fair sample assumption is best at large radii and the
Equation 2. We do this for each set of β, θ , and ∆T(0). cores of clusters may be affected significantly by physical
c
8
processes not included in our HSE model (cooling flows, HSE method. Two simulations are used, one model clus-
galaxywinds,magneticfields). Thelargestscalesonwhich ter is anoblate spheroidandthe other a prolatespheroid,
we make our calculations are determined by the shortest both with the symmetry axis in the plane of the sky. In
baselines on which we detect the SZ effect. We calculate both cases, the best fit axis ratio of the gas distribution
the statistical uncertainties in the f measurement due to was 0.79. These gas masses and gas mass fractions are
g
the shape parameter uncertainties on a number of scales, comparedto the actual model values. These are shown in
from 10′′ to 150′′. There is a broad minimum in uncer- Table 2.
tainty around radius 65′′. The model appears to be adequate at small radii, but
We calculate the gas mass fraction for the oblate and deteriorates noticeably by a radius of 100′′. This suggests
prolate spheroids at semi-major axis 65′′ ( 330h−1kpc). itis reasonableto measurethe gasmassfractionwiththis
∼
The gas mass is the same for both oblate and prolate ge- model near the same radius we constrain the data well.
ometries; the change in central density and volume when We restrict our analysis of the real observations to within
the geometry changes exactly compensate. For the oblate radius of 65′′( 330h−1 kpc), a region within which the
ellipsoid,thegasmassfractionis(0.064+0.024)h−1;forthe approximation∼is valid.
−0.024
prolateellipsoid,thegasmassfractionis(0.096+0.036)h−1.
−0.034
We also calculate f in a spherical volume of radius 65′′, 5.3. Systematic Uncertainties
g
using the fits anduncertainties fromthe sphericalfits and 5.3.1. Comparing the Lensing and HSE Masses
findf =(0.080+0.04 )h−1. Thecalculatedvaluesarecom-
g −0.041 If the assumptions made in the hydrostatic isothermal
piled in Table 1.
analysis are valid, the HSE predicted mass should equal
We also calculate the gas mass fraction for both com-
the cluster’s lensing mass. We integrate the total mass
ponents of the bimodalmodel, using a gas temperature of
density in Equation 10 in the same cylindrical volume in
6.6 keV for each and find that the gas mass fraction for
each component is 0.040h−1. which the lensing mass is calculated. This comparison is
∼ potentially a means to discriminate between oblate and
The dark matter needed to produce concentrically el-
prolate models for the cluster, but, as discussed in Sec-
lipsoidal isopotential surfaces has a significantly more as-
tion 5.2, our model for ellipsoidal clusters breaks down at
pherical distribution than the potential. For the observed
large radii. For the spherical model, this mass is calcu-
axis ratio of0.64,this darkmatter distribution is unphys-
lated for each set of shape parameters and the 68% confi-
ical beyond a few core radii in the direction of the unique
dencelimitsarederived. Usinganelectrontemperatureof
axis, as it requires the dark matter density to be nega-
6.6+1.1 keV, the spherical geometry HSE mass integrated
tive. Thisalsoimplies thatthe gasmassfractionwillvary −0.9
spatially. The difficulty of this model supporting very el- inthe40′′ radiuscylindricalvolumeis1.47+−00..4398×1014M⊙,
liptical gas density distributions may be suggesting that consistent with the lensing mass in the same geometry of
the cluster is bimodal, but it is more likely an indictment 1.59+−00..3300×1014h−1M⊙. Thetwo-componentβ-modelpre-
ofthesimplifiedmodelswehaveused. Wehaveconsidered dicts about 1.25 1014M⊙, and so it is also a consistent
×
the class of potentials with isopotential surfaces following model.
concentric ellipsoids because they adequately describe the Thisagreementdoesnotguaranteethattheassumptions
data and because their deprojections are straightforward. intheHSEanalysisarevalid,however. Forthisreason,we
Suchsimplifiedmodelsareseriouslydeficient,however,for examine possible systematic effects. Some comparisons of
the hydrostaticanalysis. Toproduceaclusterwithanob- cluster total masses derived by the HSE method to lens-
served axis ratio of less than 1/√2 in this formalism, the ing masses suggest that the HSE method systematically
cluster mass must be partially comprised of dark matter underpredicts the cluster’s mass (e.g., Miralda-Escud´e &
withnegativedensity. Itisimproper,then,tomeasurethe Babul 1994, Loeb & Mao 1994, Wu & Fang 1997). Some
darkmatterwiththismethodoutpastafewcoreradiifor of the explanations suggested for the discrepancy include
a highly elliptical cluster. clusterellipticity, non-thermalpressuresupportofthe gas
The effect of using this simple ellipsoidal model to cal- in the cluster core,multi-phase gas,and temperature gra-
culate f is assessed in the following way. We construct a dients in the gas. In an examination of a large sample of
g
modelclusterwhichhasaphysicallymotivatedmassstruc- clusters,however,Allen(1998)suggestssuchdiscrepancies
ture, predict its observed SZ effect, and then attempt to can often be resolved by taking account of cooling flows
recoverthe simulatedcluster’smassusingthe sameobser- when analyzing x-ray data, and ensuring that the lensing
vationandanalysisprotocolusedforthetrueobservations. and x-ray masses probe the same line of sight. The mass
Wearrangethedarkmatterofthesimulatedclusterincon- discrepancy does remain for clusters in the Allen sample
centrically spheroidal shells with a β-model profile. The withsmallcoolingflowsornone,perhapssuggestingthese
darkmatter’saxisratioissetto2.0:3.5. Isothermalgasat clusters have undergone recent dynamical activity which
7 keV is addedin hydrostaticequilibrium with the cluster havedisruptedanypre-existingcoolingflow. Suchactivity
potential, and a simulated two-dimensional SZ decrement might invalidate the HSE assumption.
map constructed, assuming the cluster is at redshift z = We have been careful to ensure that the strong lens-
0.37. Thedecrementmapissampledwiththeu-vcoverage ing and SZ models probe the same lines of sight. There
of a typical interferometric observation. Noise typical of is no evidence for a cooling flow in Abell 370, although
a 40 hour observation is added. The resulting u-v data there may be other concerns about using the measured
∼
are fit in the same manner described in Section 4, with emission-weighted gas temperature.
the data fit to a concentrically ellipsoidal β-model, and
the gas mass and gas mass fraction determined with the 5.3.2. Contamination of Emission-Weighted Temperature
9
Ifthemeasuredaveragetemperatureisinerror,i.e.,due value of the central density must vary inversely with the
to contamination from a nearby AGN, the mass measure- axis ratio. The gas mass, then, remains the same for all
ments will also be in error. The SZ gas mass is inversely inclination angles. This is expected, as the gas mass is
proportional to the assumed temperature and the HSE proportional to the total SZ flux, independent of its spa-
mass is directly proportionalto the temperature. The gas tialarrangement. Weevaluatethe gasandHSEmassesas
fraction from the HSE method is quite sensitive to tem- a function of i, using the best fit parameters from A370.
perature, f 1/T2. Thegasmassfractionchanges,butitchangessignificantly
g ∝ e
over a relatively small range of allowed inclination angles
5.3.3. Polytropic Temperature Gradient (see Figure 5).
Anunresolvedtemperaturegradientinthegasmaysys-
tematically affect the gas and HSE masses. If such a gra- 6. DISCUSSION
dientispresent,thetruetemperatureinthecentralregion
6.1. Gas Mass Fractions at r
maybehigherthantheemission-weightedtemperaturewe 500
use, and the fitted shape parameters from the isothermal To compare the f we have measured within a fixed
g
SZ analysis may no longer accuratelydescribe the density angular radius to f measurements in clusters with differ-
g
distribution. ent sizes and redshift, we extrapolate our measured f to
g
However,if the temperature ofthe intraclustermedium a fiducial radius. The ICM in nearby clusters has been
declines slowly, and does not change appreciably over the observed to be distributed more uniformly than the dark
angular scales to which we are sensitive, the interferomet- matter (e.g., David et al. 1995), which is to be expected
ric measurementof the gas mass fraction atthese angular if energy has been added to the intracluster medium be-
scales will in fact not be strongly affected. fore collapse or from galactic winds. If this is generally
Currently, there are no strong observationalconstraints true, the gas mass fraction measured depends on the ra-
on temperature structure in moderately distant and dis- diuswithinwhichthemeasurementismade. Assuggested
tant clusters, as there have been no suitable telescope fa- in Evrard (1997) and Metzler, Evrard, & Navarro (1998),
cilitiesforthetask. However,anattempttoquantifytem- we choose this radius to be that within which the aver-
peraturestructureobservedinnearbyclustersispresented age density of the cluster is 500 times the critical den-
in Markevitch(1996). In this work,a slow decline in tem- sity, ρ = 3H2. These numerical simulations suggest
c 8πG
perature with radius is observed, with the temperature that within this radius, r , the cluster’s baryon fraction
500
falling to half its central value at 6-10 core radii. This should closely reflect the universal baryon fraction, if the
structuremaybe approximatelydescribedbyagaswitha currentphysicalmodelsofhierarchicalstructureformation
polytropic index of γ = 1.2. (For discussion of polytropic are correct. We use the analytical expression of Evrard
indices, see Sarazin (1988) and the references within.) If (1997) which describes the expected variation of f with
g
there temperature structure in Abell 370 and it is of the overdensity. This variation is found to be consistent with
moderate variety presented in Markevitch (1996), it will thef variationreportedintheDavidetal.(1995)sample.
g
not be a strong source of systematic uncertainty.
Abell 370 has a long observation scheduled with the r (T ) η
500 e
Chandra observatory. Chandra has the necessary spatial fg(r500(Te))=fg(rX) , (11)
r
and spectral resolution to remove the effects of contam- (cid:18) X (cid:19)
inating sources in the field and with a long observation, where η = 0.17, f (r (T )) is the gas mass fraction at
g 500 e
could measure the spatial variation of the cluster temper- r , and r is the radius within which the gas mass frac-
500 X
ature, should it exist. tionis measured. We modify Evrard’sexpressionfor r ,
500
derived for low redshift clusters, to include the change in
5.3.4. Inclination Angle the value of ρc with redshift; ρc(z) = ρc(z = 0)(H/H◦)2,
whereH2 =H2[(1+z)3Ω +(1+z)2(1 Ω Ω )+Ω ]
The assumption that the cluster is biaxially symmetric ◦ M M Λ Λ
− −
with symmetry axis in the plane of the sky is certainly
1/2
a simplification. Here, we estimate the effect on the gas r (T )=(1.24 0.09) Te h−1Mpc.
mass fraction of an inclined symmetry axis. Analytical 500 e ± 10 keV(H/H◦)2
(cid:18) (cid:19)
relationshipsbetweentheinclinationangleandtheappar- (12)
ent shape of an biaxial ellipsoid are derived in Fabricant, The gas mass fraction values at r , estimated from
500
Rybicki, & Gorenstein (1984). We derive the cluster gas those measured at 65′′, are summarized in Table 1.
mass and HSE mass for a cluster which reproduces the This experiment best measures f at a given angular
g
observed SZ map, but has an inclined symmetry axis. A scale,which correspondsin Abell 370to an overdensityof
biaxialcluster,wheninclined,willretainaβ-modeldistri- 5000ρ . Thisisnottheoptimalradiusatwhichtocom-
c
∼
bution with the same value of β, but the central density parewithnumericalsimulations,sinceresolutionislimited
and intrinsic axis ratio will change. The inclination angle in the cores of the clusters, and the gas in the core may
i is measured between the symmetry axis and the line of also be sensitive to additional physics not yet included in
sight; i=90◦ for the analysis of Section 5.2. the models, e.g., magnetic fields and cooling. For these
As inclination angle increases, so does the intrinsic axis reasons, the corrections should be taken with some cau-
ratio.7 In order to preserve the observed SZ effect, The tion.
7Therelationshipbetween intrinsicaxis ratio,ai,andthe inclinationangle, i,foranoblate ellipsoidisai =a[1−a−2sicno(si2)(i)]1/2,whereais
theobservedaxisratio. Foraprolateellipsoid,ai=a[1−a2csoins2i(i)]1/2.
10
6.2. Constraints on Ω from f fromthegravitationallensingmodelforboththeone-and
M g
two-component models, lending support to the HSE as-
Underthe fairsamplehypothesis,A370’sgasmassfrac-
sumption. The surface gas mass fraction measurement is
tion within r , a lower limit to the cluster baryon frac-
500 made within an angular radius of 40′′ and the HSE gas
tion, should reflect the universal baryon fraction:
massfractionismadewithinaradiusof65′′. Thegasmass
ΩB fractionnearthevirialradiusisderivedfromthegasmass
f f = , (13)
g ≤ B Ω fractionsat65′′ using a correctionfactorderivedfromnu-
M
mericalsimulations. Fortherangeofmethodsandmodels
where f is the cluster’s baryon fraction, Ω is the ra-
B M used, we find gas mass fraction values of (5 13)h−1%.
tio of the total mass density to the critical mass density,
∼ −
andΩ istheratioofbaryonmassdensityintheuniverse Constraints on the Hubble parameter H◦ can in princi-
B
ple be derived from the SZ and x-ray measurements of a
to the critical mass density. The cluster gas mass frac-
cluster. The SZ and x-ray observables depend on differ-
tion measurements can then be used within the Big Bang
ent moments of the electron density, and so the charac-
Nucleosynthesis (BBN) paradigm to constrain Ω :
M
teristic length scale of the cluster along the line of sight
ΩM ΩB/fg. (14) can be measured and the angular diameter distance of
≤
The value of Ω is constrained by BBN calculations the cluster inferred. This is useful not only as a distance
B
and the measurements of light element abundances. The ladder-independent measurement of H◦, but, when com-
relative abundance of deuterium and hydrogen provides paredwithotherH◦ measurements,canbeusedtoexplore
a particularly strong constraint on the baryonic matter possible systematic effects in the fg calculation, e.g., to
density. A firm upper limit to Ω is set by the presence constrain the deprojection of the gas distribution. How-
B
of deuterium in the local interstellar medium. This con- ever,thequalityofthex-rayimagingdatainthiscaseand
strainsthevalueofΩ tobelessthan0.031h−2(Linskyet the apparent disagreement between the SZ and x-ray fit-
B
al. 1995). Measurements of the D/H ratio in metal-poor ted models do not permit putting a strong constraint on
Lyman-α absorption line systems in high-redshift quasars the Hubble constant. A long observation with the Chan-
putatighterconstraintonthebaryonicmassdensity. Such dra x-rayobservatorytowardsthis cluster is planned, and
measurements made by Burles & Tytler (1998) predict a should help resolve these issues.
value of Ω =(0.019 0.002)h−2 at 95% confidence. The valueofthebaryonicmassfractioninanyoneclus-
B
The gas mass fract±ions measured for A370 from both ter will be susceptible to systematic uncertainties which
lensing and HSE methods range from 5 13%h−1. We may be difficult to estimate. In order to use cluster gas
consider the simplest f measurement, th−at in the spher- mass fractions as a cosmological tool, one wants to ame-
g
ical model, and compare this gas mass fraction at r lioratetheeffectoftheseerrorsbystudyingalargesample
500
to the Burles & Tytler (1998) value for Ω . This gives of clusters. A SZ effect survey in galaxy clusters is being
B
an upper limit to the matter density parameter, Ω carried out by this group at the BIMA and OVRO obser-
M
0.19+0.10h−1, at 68% confidence. However, the bimod≤al vatories, and an analysis of the gas mass fraction of the
−0.10
model gives a surface gas mass fraction at angular radius sample is in preparation.
40′′ of 0.048h−1, a value which permits Ω to be as high Many thanks are due to the staff at the BIMA and
M
as 0.40h−1 in this scheme. (We note again that the de- OVROobservatoriesfortheircontributionstothisproject,
pendence of f through the angular diameter distance is especially Rick Forster, John Lugten, Steve Padin, Dick
g
weak at this redshift, with a change in f of 5-10% when Plambeck, Steve Scott, and Dave Woody. Many thanks
g
a wide range of cosmological parameters is used.) These to Cheryl Alexander for her work on the system hard-
values are consistent with the limits on Ω from obser- ware. Thanks also to Jack Hughes and Doris Neumann
M
vations ofsupernovae,whichare derivedfromgeometrical for valuable discussions concerning the x-ray analysis, to
arguments, rather than the Ω /f ratio. Depending on Jean-Paul Kneib concerning his lensing analysis, and to
B g
the method used to calibrate the sample, for a spatially Naomi Ota and collaborators for sharing their reduced
flat universe, Garnavich et al. (1998) find Ω <0.4 0.5 ASCAdata. ThisworkissupportedbyNASALTSAgrant
M
at 68% confidence. − NAG5-7986. LG, EDR, and SKP gratefully thank the
NASA GSRP program for its support. Radio astronomy
withtheOVROandBIMAmillimeter arraysis supported
6.3. Conclusion & Future Work
by NSF grant AST 93-14079 and AST 96-13998, respec-
We have measured the Sunyaev-Zel’dovicheffect in the tively. The funds for the additional hardware for the SZ
galaxy cluster Abell 370 and present spatially filtered im- experiment were from a NASA CDDF grant, a NSF-YI
ages from these data. The optical and x-ray observations Award, and the David and Lucile Packard Foundation.
of this cluster show a complicated and perhaps bimodal
mass distribution. The SZ effect image, however, looks
smoothlydistributedandsignificantlyaspherical. Wehave
fit both one-component and two-component β-models to
the dataandfindthatthe two-componentmodeldoesnot
fit significantly better. For both models, we calculate the
gasmassfractionfortheclusterusingmeasurementsofthe
total cluster mass from both the gravitational lens model
(the“surface”gasmassfraction)andfromthehydrostatic
equilibriumassumption. Whenintegratedinthesamevol-
ume,theHSEmassesareconsistentwiththemassderived